The Veronese Surface in PG(5,3) and the Small Witt Design

The picture on the left hand side shows the projective plane over GF(3), i.e. the field of residue classes modulo three. This plane has thirteen points (the coloured polygons) and thirteen lines (the black curves). Each point is on exactly four lines and each line carries exactly four points.

The four green squares comprise a conic in that plane. It has four tangent lines (meeting the conic in one point only), three interior lines (missing the conic), and six external lines (meeting the conic in exactly two points). Dually, there are three internal points (lying on no tangent) and six external points (lying on exactly two tangents). Those points are illustrated by red triangles and blue hexagons, respectively. It is a peculiar property, that the green conic and the red triangle determine each other uniquely.

This property is the backbone of some of our results on the relationship between the Veronese surface in the 5-dimensional space over GF(3) and a 12-cap in that space which is a point model for the Witt designW12, a combinatorial structure with twelve points and 132 blocks. As a matter of fact, this 12-cap has been found before by H.S.M. Coxeter using a completely different approach.